Katya is a ranger at a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population over time. The relationship between the elapsed time $t$, in years, since the beginning of the study and the bear population $B(t)$, on the reserve is modeled by the following function. $B(t)=5000 \cdot 2^{-0.05t}$ In how many years will the reserve's bear population be $2000$ ? Round your answer, if necessary, to the nearest hundredth.
Thinking about the problem We want to know how many years, $t$, it will take for the bear population, $B(t)$, to reach $2000$. So we need to find the value of $t$ for which $B(t)=2000$. Substituting $2000$ in for $B(t)$ in the function gives us the following equation. $2000=5000 \cdot 2^{-0.05t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}5000\cdot2^{-0.05t}&=2000\\\\ 2^{-0.05t}&=0.4\\\\ -0.05t&=\log_2(0.4)\\\\ t&=\dfrac{\log_2(0.4)}{-0.05}\\\\ \end{aligned}$ Changing the base to approximate the solution Since most calculators only calculate logarithms in base $10$ and base $e$, let's change the base. [What is the change of base rule?] $\begin{aligned}t&=\dfrac{\log_2(0.4)}{-0.05}\\\\ &=\dfrac{1}{-0.05}\cdot \dfrac{\log(0.4)}{\log(2)}\\\\ &\approx 26.44 \end{aligned}$ The bear population in the reserve will be at $2000$ bears after $26.44$ years.